Control devices exercise a restraining or directing influence over a controllable device. During its operation an automatic control device typically receives one or more input parameters and in response to the received input parameters outputs one or more control functions. Control functions are further input to a controllable device and they bring about an effect on the operation of the controllable device. One control device may control several controllable devices and one controllable device may be controlled by one or more control devices. The input parameters of the control device may originate from a process of which the controllable device is a part, or they may be fed into the control process from outside, for example as control commands or as output parameters from another process.
In automatic control devices the input parameters are typically derived from a series of sampled data. The document NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) copyright © 1988-1992 by Cambridge University Press, pp. 496-510, provides the basic principles of Fast Fourier Transform and the referred pages are incorporated herein by reference.
The computation is based on the perception that a physical process can be described in the time domain by the values of some quantity h as a function of time t, or in the frequency domain where the process is specified by giving its amplitude H as a function of frequency f. These two representations can be related to another by means of the Fourier transform equations:
                                                                        H                ⁡                                  (                  f                  )                                            =                                                ∫                                      -                    ∞                                    ∞                                ⁢                                                      h                    ⁡                                          (                      t                      )                                                        ⁢                                      ⅇ                                          2                      ⁢                      π                      ⁢                                                                                          ⁢                      ift                                                        ⁢                                      ⅆ                    t                                                                                                                                          h                ⁡                                  (                  t                  )                                            =                                                ∫                                      -                    ∞                                    ∞                                ⁢                                                      H                    ⁡                                          (                      f                      )                                                        ⁢                                      ⅇ                                                                  -                        2                                            ⁢                      π                      ⁢                                                                                          ⁢                      ift                                                        ⁢                                      ⅆ                    f                                                                                                          (        1        )            
In the most typical situations, function h(t) is sampled at evenly spaced intervals in time, so that the sequence of n sampled values hn ishn=h(nΔ) n= . . . , −3, −2, −1, 0.1, 2, 3, . . .   (2)where Δ is the sampling rate. The integral of equation (1) can be approximated by a discrete sum
                                                                        H                ⁡                                  (                                      f                    n                                    )                                            =                            ⁢                                                ∫                                      -                    ∞                                    ∞                                ⁢                                                      h                    ⁡                                          (                      t                      )                                                        ⁢                                      ⅇ                                          2                      ⁢                      π                      ⁢                                                                                          ⁢                      ⅈ                      ⁢                                                                                          ⁢                                              f                        n                                            ⁢                      t                                                        ⁢                                                                          ⁢                                      ⅆ                    t                                                                                                                          ≈                            ⁢                                                ∑                                      k                    =                    0                                                        N                    -                    1                                                  ⁢                                                      h                    k                                    ⁢                                      ⅇ                                          2                      ⁢                      π                      ⁢                                                                                          ⁢                      ⅈ                      ⁢                                                                                          ⁢                                              f                        n                                            ⁢                                              t                        k                                                                              ⁢                  Δ                                                                                                        =                            ⁢                              Δ                ⁢                                                      ∑                                          k                      =                      0                                                              N                      -                      1                                                        ⁢                                                            h                      k                                        ⁢                                          ⅇ                                              2                        ⁢                        π                        ⁢                                                                                                  ⁢                        ⅈ                        ⁢                                                                                                  ⁢                                                  kn                          /                          N                                                                                                                                                                            (        3        )            wherebyH(fn)≈ΔHn  (4)when
                              H          n                ≡                              ∑                          k              =              0                                      N              -              1                                ⁢                                    h              k                        ⁢                          ⅇ                              2                ⁢                πⅈ                ⁢                                                                  ⁢                                  kn                  /                  N                                                                                        (        5        )            
In practical solutions the computing related to Discrete Fourier Transform is significantly reduced by utilising Fast Fourier Transform (FFT) algorithm. In the first section of FFT, the data is first sorted into bit-reversed order. The second section of FFT has an outer loop that is executed log2 N times and calculates, in turn, transform of length 2, 4, . . . , N. For each stage of the process the two nested inner loops range over the sub-transforms already computed and the elements of each transform, implementing the Danielson-Lanczos Lemma. This variant of the FFT is called a decimation-in-time or Cooley-Tukey FFT algorithm. The decimation-in-frequency (Sande-Tukey) FFT algorithm first goes through a set of log2N iterations on the input data, and then rearranges the output values into bit-reverse order.
In another class of FFT variants the initial data set of N is sub-divided down to some small power of 2, for example N=4 (base-4 FFT) or N=8 (base-8 FFT). These small transforms are done by small sections of optimized coding that utilize special symmetries of the particular N. Assuming, for example, that N=4, leads to the trigonometric sines and cosines being 0, +1 or −1, which by far eliminates multiplications, and leaves additions and subtractions. Base-4 or base-8 FFTs have been considered to decrease the computing of the order to 20 to 30 percent.
One example of automatic control devices is an electric protection device. The general purpose of electric protection devices is to prevent or limit damage to the protected device or to protect service of the protected device from interruption. In electric circuits the current flows are rapidly varied and the power may surge instantaneously. Therefore the power feed of an electrically driven device is always provided with one or more protection functions that are capable of isolating the device from harmful electrical transients from the power feed. In more elementary devices the protection devices are typically relay arrangements that are triggered when present current or voltage levels exceed a pre-determined threshold. In more complex systems the current and voltage levels are rigorously monitored and protection functions responsive to various parameters acquired through monitoring are utilized to safeguard the system from harmful effects in the power system.
In general electrical devices are designed to function with sinusoidal voltage, and for accurate functionality highly developed systems also require a faultless voltage to function correctly. One of the main harmful effects deteriorating the quality of electricity in the power systems are the harmonics. Harmonics are divided into different components by their properties (ranking number). The component class shows the rotational direction of the phase phasor of the harmonic with respect to the fundamental frequency.
When the curve form of the voltage or of the current deviates from the sinusoidal, it can be considered formed of several sinusoidal signals of different frequency. The function is typically broken down into its components by using the above Fourier analysis, on which the mathematical treatment of harmonics is based.
In complex protection devices the method of determining a protection function based on said input parameters generally comprises balancing between the expenses of the device and the response time from receiving the input parameters to outputting the determined protection function. In known protection devices a digital signal processor (DSP) is considered the most powerful tool for handling defined input signals and for determining a protection function thereupon. However, the component costs related to digital signal processors are considerable, and in a complex system the costs due to the number of digital signal processors need to be optimised.
On the other hand, the embedded systems facilitate parallel usage of a variety of programmable algorithms. A number of variable algorithms are available and component costs are not directly increased with the number of operable protection functions. However, going through the number of calculations takes time and the accumulated response times become longer than what is actually required in order to implement the protection functions appropriately. There are a number of cases where the acceptable response times are only slightly exceeded, but for reliability reasons the costly DSP very often end up being preferred over the programmable algorithms.
Consequently, one of the disadvantages associated with the known electric protection devices is that the available means for determining protection functions in response to various input values do not facilitate adequate optimization that takes into consideration both the technical and economical factors related to operable protection functions.